IODS course project
Antero Salminen
- Introduction to Open Data Science - Course Project
- About the project
- Chapter 1
- Chapter 2: Assignment 2: Analysis (max 15 points)
- 1. Read the data & describe the dataset briefly
- 2. Graphical overview
- 3. Regression model
- 4. explaining the relationship between explanatory variables and the target variable
- 5. Diagnostic plots
- Assignment 4. Analysis exercises (Max 15 points)
Introduction to Open Data Science - Course Project
About the project
Write a short description about the course and add a link to your GitHub repository here. This is an R Markdown (.Rmd) file so you should use R Markdown syntax.
This is course diary of the course Introduction to Open Data Science. My repository can be found here: https://github.com/anterogradinen/IODS-project
# This is a so-called "R chunk" where you can write R code.
date()## [1] "Mon Nov 27 22:08:33 2023"Chapter 1
Assignment 1: Tasks and Instructions copied from Moodle.
DONE. Check that you have everything installed and created according to the instructions. You should have a GitHub repository, a course diary web page (also on GitHub, in a different address) and the IODS-project started on RStudio using the course templates. (3 p)
DONE. Open the file chapter1.Rmd located in your IODS-project folder with RStudio. Just write some of your thoughts about this course freely in the file, e.g., How are you feeling right now? What do you expect to learn? Where did you hear about the course?
Feelings. I am excited of this course! I am already with one R-course but this course takes in consideration open data aspect which is really interesting and future skill.
Expectations. I really hope that this and other R-course I have support my learning process to become starting data-analyst. Also the aspect of open data and its possibilities really inspires me.
Where did I hear about this course? I have came across this course coupe of times but now I really have the possibility to participate to this.
Also reflect on your learning experiences with the R for Health Data Science book and the Exercise Set 1:
How did it work as a “crash course” on modern R tools and using RStudio?
I found the book very useful, because it introduced R and RStudio bit different than book “R for Data Science (2e)” by Wickham and Çetinkaya-Rundel (https://r4ds.hadley.nz/), which I have read previously during this autumn. Both books support each other.
Although I felt that first five chapters in a one week was bit too much. (I am currently writing this after chapter 3.5.)
First I wasn’t sure if the Excercise1-material was needed because the book provided the code which can be copy-pasted, but the exercise material saved lot of time when you did not have to download all the example data and copy-paste every example script.
Which were your favorite topics?
- I have found chapters 3.3., 3.4. and 3.5. so far most useful. (I am currently writing this after chapter 3.5.)
Which topics were most difficult?
- I think that that internalizing how to use group_by and summarise may take some time. Also ungrouping was something I need to return later on.
Some other comments on the book and our new approach of getting started with R Markdown etc.? (All this is just “warmup” to get well started and learn also the technical steps needed each week in Moodle, that is, submit and review.
We will start more serious work next week! You can already look at the next topic in Moodle and begin working with the Exercise Set 2...)
DONE Also add in the file a link to your GitHub repository (that you created earlier): https://github.com/anterogradinen/IODS-project
You can immediately start to learn the basics of the R Markdown syntax that we will use for writing the exercise reports: Try, for example, highlighting parts of your text, adding some headers, lists, links etc. Hint: Use the R Markdown Reference Guide or cheatsheet (both found from the RStudio Help). This is an excellent quick (1 min) tour of R Markdown, please watch: https://rmarkdown.rstudio.com/lesson-1.html
DONE. Remember to save your chapter1.Rmd file. (5 p)
DONE Open the index.Rmd file with RStudio. At the beginning of the file, in the YAML options below the ‘title’ option, add the following option: author: “Your Name”. Save the file and “knit” the document (there’s a button for that) as an HTML page. This will also update the index.html file. (2 p)
DONE. (This point added in 2022 - let’s hope it works similarly in 2023!)
To make the connection between RStudioand GitHub as smooth as possible, you should create a Personal Access Token (PAT).
The shortest way to proceed is to follow the steps below. (Source: https://happygitwithr.com/https-pat.html)
Execute these R commands in the RStudio Console (below the Editor):
install.packages("usethis") usethis::create_github_token()GitHub website will open in your browser. Log in with your GitHub credentials.
Write a Note in the box, for example “IODS Project”.
Select an Expiration time for your PAT, e.g., 50 days.
The pre-selected scopes “repo”, “workflow”, “gist”, and “user” are OK.
Press “Generate token” and copy the generated PAT to your clipboard. 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
Return to RStudio and continue in the Console:
gitcreds::gitcreds_set()WAIT until a prompt “Enter password or token:” appears.
Paste your PAT to the prompt and press Enter.
Now you should be able to work with GitHub, i.e., push and pull from RStudio. Congrats!! (5 p)
Upload the changes to GitHub (the version control platform) from RStudio.
There are a few phases (don’t worry: all this will become an easy routine for you very soon!):
DEMO
First, select the “Git” tab in the upper right corner of RStudio. You will see a list of modified files.
Select “Commit”. It will open a new “Review Changes” window showing more detailed information of the changes you have made in each file since the previous version.
Tick the box in the front of each file (be patient, it takes some time for the check to appear).
Write a small commit message (there’s a box for that) that describes your changes briefly. After this task is completed (not yet), both the changes and the message will be seen on GitHub. (Note: It is useful to make commits often and even on small changes. Commits are at the heart of the version control system, as a single commit represents a single version of the file.)
Press “Commit”. (RStudio uses Git to implement the changes included in the commit.)
Press “Push”. (RStudio uses Git to upload the changes to your GitHub repository.)
Now you can close the “Review Changes” window of RStudio. Good job!! (5 p)
After a few moments, go to your GitHub repository at
https://github.com/anterogradinen/IODS-project
to see what has changed (please be patient and refresh the page).
Also visit your course diary that has been automatically been updated at
https://anterogradinen.github.io/IODS-project/ and make sure you see the changes there as well.
After completing the tasks above you are ready to submit your
Assignment for the review (using the Moodle Workshop below).
Have the two links (your GitHub repository and your course
diary) ready! Remember to get back there when the Review phase begins
(see course schedule).
Have fun and don’t be afraid to ask for help using the Moodle discussion forum.
Chapter 2: Assignment 2: Analysis (max 15 points)
This weeks assignment was a tough one! Also I had last minute tech issues with knitting. Apologies if my code and text is difficult to read. I tried to use code chunks (like in the “date()” -part) in parts 1-5 but could not knit whole thing so the code is there in a regular text.
First I read the material and then started doing exercises and only then doing Assignment 2 itself. I feel that this was not the most time efficient way to learn. But this weeks learning curve was quite steep! I think I have learned quite lot about linear models this week. Also I found the data and data wrangling exercise really useful regarding my own research topic.
Nevertheless I think I need to read the material again because I am not 100% confident when looking model summaries and tables. I also fear that if I do not internalise these topics well enough, the rest of the course will be torment or I can drop out.
date()## [1] "Mon Nov 27 22:08:33 2023"1. Read the data & describe the dataset briefly
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(finalfit)
students2014_data <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/Helsinki-Open-Data-Science/master/datasets/learning2014.txt", sep = ",", header = TRUE)
View(students2014_data)
# describe the dataset briefly
glimpse(students2014_data)## Rows: 166
## Columns: 7
## $ gender <chr> "F", "M", "F", "M", "M", "F", "M", "F", "M", "F", "M", "F", "…
## $ age <int> 53, 55, 49, 53, 49, 38, 50, 37, 37, 42, 37, 34, 34, 34, 35, 3…
## $ attitude <dbl> 3.7, 3.1, 2.5, 3.5, 3.7, 3.8, 3.5, 2.9, 3.8, 2.1, 3.9, 3.8, 2…
## $ deep <dbl> 3.583333, 2.916667, 3.500000, 3.500000, 3.666667, 4.750000, 3…
## $ stra <dbl> 3.375, 2.750, 3.625, 3.125, 3.625, 3.625, 2.250, 4.000, 4.250…
## $ surf <dbl> 2.583333, 3.166667, 2.250000, 2.250000, 2.833333, 2.416667, 1…
## $ points <int> 25, 12, 24, 10, 22, 21, 21, 31, 24, 26, 31, 31, 23, 25, 21, 3…# 166 respondents: 166 rows and 7 columns
ff_glimpse(students2014_data)## $Continuous
## label var_type n missing_n missing_percent mean sd min
## age age <int> 166 0 0.0 25.5 7.8 17.0
## attitude attitude <dbl> 166 0 0.0 3.1 0.7 1.4
## deep deep <dbl> 166 0 0.0 3.7 0.6 1.6
## stra stra <dbl> 166 0 0.0 3.1 0.8 1.2
## surf surf <dbl> 166 0 0.0 2.8 0.5 1.6
## points points <int> 166 0 0.0 22.7 5.9 7.0
## quartile_25 median quartile_75 max
## age 21.0 22.0 27.0 55.0
## attitude 2.6 3.2 3.7 5.0
## deep 3.3 3.7 4.1 4.9
## stra 2.6 3.2 3.6 5.0
## surf 2.4 2.8 3.2 4.3
## points 19.0 23.0 27.8 33.0
##
## $Categorical
## label var_type n missing_n missing_percent levels_n levels
## gender gender <chr> 166 0 0.0 2 -
## levels_count levels_percent
## gender - -Comments
- age median is 22.0 and mean is 25.5.
- youngest 17 year old and oldest 55 year old,
- points mean 22.7, 23 median, min 7.0 and max 33.00.
Other variables:
- attitude (mean 3.1),
- deep approach (mean 3.7),
- stategic approach (mean 3.1) and
- surface approach (mean 2.8)
students2014_data |> count(gender)
- gender split: 110 female (66.3%), 56 male (33.7%)
2. Graphical overview
Show a graphical overview of the data and show summaries of the variables in the data. Describe and interpret the outputs, commenting on the distributions of the variables and the relationships between them.
library(GGally) ## Loading required package: ggplot2## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2library(ggplot2)
# "create a plot matrix with ggpairs()"
p <- ggpairs(students2014_data, mapping = aes(col = gender, alpha = 0.3), lower = list(combo = wrap("facethist", bins = 20)))
# "draw the plot"
p
library(GGally) library(ggplot2)
“create a plot matrix with ggpairs()”
p <- ggpairs(students2014_data, mapping = aes(col = gender, alpha = 0.3), lower = list(combo = wrap(“facethist”, bins = 20)))
“draw the plot”
p
summaries about the variables stated above in part 1.
Comments based on graphical overview.
As mentioned above: more female respondents than male, but male respondents bit more older.
Attitude towards statistics: male have more positive attitude than female towards statistics in this data. - General observation about learning approaches: males responded having tiny bit more deep approach than female, female responded having tiny bit more strategic and surface approaches than male.
When looking correlation matrix noteworthy are that attitudes and points are positively correlated in this data and surface and deep approaches are negatively correlated.
3. Regression model
Choose three variables as explanatory variables and fit a regression model where exam points is the target (dependent, outcome) variable. Show a summary of the fitted model and comment and interpret the results.
Explain and interpret the statistical test related to the model parameters. If an explanatory variable in your model does not have a statistically significant relationship with the target variable, remove the variable from the model and fit the model again without it. (0-4 points)
Explanatory variables (selected based on correlation matrix): 1) Global attitude toward statistics, 2) gender and 3) strategic approach to learning.
Dependent, outcome: Exam points
create a regression model with multiple explanatory variables
my_model3 <- lm(points ~ attitude + stra + gender, data = students2014_data)
“print out a summary of the model”
summary(my_model3)
Explain and interpret the statistical test related to the model parameters.
Attitude has statistically significant connection to points. In this model it’s slope is 3.5. Meaning that positive attitude to statistics had connection to higher points (one point increase in attitude average increased 3.5 of total points during course).
Strategic approach had positive connection to points, but statistically this connection was not significant.
Males had negative connection to points, but statistically this connection was not significant.
Adjusted R-squared is low (0.1904), meaning that the model does not describe data well.
Trying to make better model: If an explanatory variable in your model does not have a statistically significant relationship with the target variable, remove the variable from the model and fit the model again without it.
my_model4 <- lm(points ~ attitude, data = students2014_data) summary(my_model4)
Comments about my_model4:
- no mentionable changes in model. Adjusted R-squared is still low (0.1804)
4. explaining the relationship between explanatory variables and the target variable
“Using a summary of your fitted model, explain the relationship between the chosen explanatory variables and the target variable (interpret the model parameters).”
Explain and interpret the multiple R-squared of the model. (0-3 points)
qplot(attitude, points, data = students2014_data) + geom_smooth(method = “lm”)
summary(my_model3)
Explain and interpret the multiple R-squared of the model.
Based on this model: 20% of the variation of points are explained by these selected variables
I also add my observation from task 3 above:
Attitude has statistically significant connection to points. In this model it’s slope is 3.5. Meaning that positive attitude to statistics had connection to higher points (one point increase in attitude average increased 3.5 of total points during course).
Strategic approach had positive connection to points, but statistically this connection was not significant.
Males had negative connection to points, but statistically this connection was not significant.
Adjusted R-squared is low (0.1904), meaning that the model does not describe data well.
5. Diagnostic plots
Produce the following diagnostic plots: Residuals vs Fitted values, Normal QQ-plot and Residuals vs Leverage.
Explain the assumptions of the model and interpret the validity of those assumptions based on the diagnostic plots. (0-3 points)
my_model3 <- lm(points ~ attitude + gender, data = students2014_data) plot(my_model3, which = c(1,2,5))
Explination and interpretation: - The observations are normally distributed around the fitted line because a normal Q-Q plot shows residuals in line with the straight line. - this is good sign, meaning that residuals are observations are equally distributed aroung the linear model line, which can be also visually seen in qplot above.
Assignment 3
Analysis
date() #testing does the code chunk work (had issues with assignment 2)## [1] "Mon Nov 27 22:08:48 2023"As mentioned above, I had some tech issues with last knitting and code chunk. As I did with last week’s assignment, I did have had difficult time also with this weeks material and assignment. Unfortunately I did not manage to do every task required. I am bit worried will I fall behind in this course. Nevertheless I found the exercise material really interesting.
2
Read the joined student alcohol consumption data into R either from your local folder (if you completed the Data wrangling part) or from this url (in case you got stuck with the Data wrangling part):
https://raw.githubusercontent.com/KimmoVehkalahti/Helsinki-Open-Data-Science/master/datasets/alc.csv
(In the above linked file, the column separator is a comma and the first row includes the column names). Print out the names of the variables in the data and describe the data set briefly, assuming the reader has no previous knowledge of it. There is information related to the data here. (0-1 point)
library(dplyr)
library(tidyr)
library(finalfit)
library(tidyverse)## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ forcats 1.0.0 ✔ readr 2.1.4
## ✔ lubridate 1.9.2 ✔ stringr 1.5.0
## ✔ purrr 1.0.2 ✔ tibble 3.2.1
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorsalc3 <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/Helsinki-Open-Data-Science/master/datasets/alc.csv", sep = ",", header = T)
glimpse(alc3)## Rows: 370
## Columns: 35
## $ school <chr> "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP",…
## $ sex <chr> "F", "F", "F", "F", "F", "M", "M", "F", "M", "M", "F", "F",…
## $ age <int> 18, 17, 15, 15, 16, 16, 16, 17, 15, 15, 15, 15, 15, 15, 15,…
## $ address <chr> "U", "U", "U", "U", "U", "U", "U", "U", "U", "U", "U", "U",…
## $ famsize <chr> "GT3", "GT3", "LE3", "GT3", "GT3", "LE3", "LE3", "GT3", "LE…
## $ Pstatus <chr> "A", "T", "T", "T", "T", "T", "T", "A", "A", "T", "T", "T",…
## $ Medu <int> 4, 1, 1, 4, 3, 4, 2, 4, 3, 3, 4, 2, 4, 4, 2, 4, 4, 3, 3, 4,…
## $ Fedu <int> 4, 1, 1, 2, 3, 3, 2, 4, 2, 4, 4, 1, 4, 3, 2, 4, 4, 3, 2, 3,…
## $ Mjob <chr> "at_home", "at_home", "at_home", "health", "other", "servic…
## $ Fjob <chr> "teacher", "other", "other", "services", "other", "other", …
## $ reason <chr> "course", "course", "other", "home", "home", "reputation", …
## $ guardian <chr> "mother", "father", "mother", "mother", "father", "mother",…
## $ traveltime <int> 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1,…
## $ studytime <int> 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 1, 3, 2, 1, 1,…
## $ schoolsup <chr> "yes", "no", "yes", "no", "no", "no", "no", "yes", "no", "n…
## $ famsup <chr> "no", "yes", "no", "yes", "yes", "yes", "no", "yes", "yes",…
## $ activities <chr> "no", "no", "no", "yes", "no", "yes", "no", "no", "no", "ye…
## $ nursery <chr> "yes", "no", "yes", "yes", "yes", "yes", "yes", "yes", "yes…
## $ higher <chr> "yes", "yes", "yes", "yes", "yes", "yes", "yes", "yes", "ye…
## $ internet <chr> "no", "yes", "yes", "yes", "no", "yes", "yes", "no", "yes",…
## $ romantic <chr> "no", "no", "no", "yes", "no", "no", "no", "no", "no", "no"…
## $ famrel <int> 4, 5, 4, 3, 4, 5, 4, 4, 4, 5, 3, 5, 4, 5, 4, 4, 3, 5, 5, 3,…
## $ freetime <int> 3, 3, 3, 2, 3, 4, 4, 1, 2, 5, 3, 2, 3, 4, 5, 4, 2, 3, 5, 1,…
## $ goout <int> 4, 3, 2, 2, 2, 2, 4, 4, 2, 1, 3, 2, 3, 3, 2, 4, 3, 2, 5, 3,…
## $ Dalc <int> 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1,…
## $ Walc <int> 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, 1, 4, 3,…
## $ health <int> 3, 3, 3, 5, 5, 5, 3, 1, 1, 5, 2, 4, 5, 3, 3, 2, 2, 4, 5, 5,…
## $ failures <int> 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0,…
## $ paid <chr> "no", "no", "yes", "yes", "yes", "yes", "no", "no", "yes", …
## $ absences <int> 5, 3, 8, 1, 2, 8, 0, 4, 0, 0, 1, 2, 1, 1, 0, 5, 8, 3, 9, 5,…
## $ G1 <int> 2, 7, 10, 14, 8, 14, 12, 8, 16, 13, 12, 10, 13, 11, 14, 16,…
## $ G2 <int> 8, 8, 10, 14, 12, 14, 12, 9, 17, 14, 11, 12, 14, 11, 15, 16…
## $ G3 <int> 8, 8, 11, 14, 12, 14, 12, 10, 18, 14, 12, 12, 13, 12, 16, 1…
## $ alc_use <dbl> 1.0, 1.0, 2.5, 1.0, 1.5, 1.5, 1.0, 1.0, 1.0, 1.0, 1.5, 1.0,…
## $ high_use <lgl> FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE, FALS…Description:
3
The purpose of your analysis is to study the relationships between high/low alcohol consumption and some of the other variables in the data. To do this, choose 4 interesting variables in the data and for each of them, present your personal hypothesis about their relationships with alcohol consumption. (0-1 point)
For this excercise I study relationship between high and low alcohol consumption gender, education level of parents, and motivation to take higher education.
I am interested to look wheter there is relationship between student’s motivation to take higher education and education level of parents to alcohol consumption.
Variables are following (described in: http://www.archive.ics.uci.edu/dataset/320/student+performance)
7 Medu - mother’s education (numeric: 0 - none, 1 - primary education (4th grade), 2 — 5th to 9th grade, 3 — secondary education or 4 — higher education)
8 Fedu - father’s education (numeric: 0 - none, 1 - primary education (4th grade), 2 — 5th to 9th grade, 3 — secondary education or 4 — higher education)
17 famsup - family educational support (binary: yes or no)
21 higher - wants to take higher education (binary: yes or no)
My personal hypothesis about these variables is that would have low (if any) negative correlation:the higher family’s education level and student’s “education motivation” the lower alchohol consumption.
I also computed mean education level based on both parents education level.
4
Numerically and graphically explore the distributions of your chosen variables and their relationships with alcohol consumption (use for example cross-tabulations, bar plots and box plots).
Comment on your findings and compare the results of your exploration to your previously stated hypotheses. (0-5 points)
First we look how much alcohol is used in different genders. We see that data have 70 male students with high use of alcohol and female students 41.
While looking boxplots of parents’ education and high use of alcohol consumption, I do not see much anything of noteworthy. Father’s education level median is bit lower with male students with high alcohol consumption. But father’s education level median was also lower with female student with low alcohol consumption. I think that boxplot is not the best graphic with this data.
This observations do not really support my hypothesis (but do not refute it necessarily). I also note that would analyze both parents’ education level and alcohol consumption bit better than with mean grade. Unfortunately I ran out of time with this assignment. :(
### bar plots ###
# A plot of alcohol use with gender
alc3 |>
group_by(sex) |>
count(alc3$high_use)## # A tibble: 4 × 3
## # Groups: sex [2]
## sex `alc3$high_use` n
## <chr> <lgl> <int>
## 1 F FALSE 154
## 2 F TRUE 41
## 3 M FALSE 105
## 4 M TRUE 70g1 <- ggplot(data = alc3, aes(x = high_use))
g1 + geom_bar() + facet_wrap("sex")
### box plots ###
# a plot of high_use and mother's education
g1 <- ggplot(alc3, aes(x = high_use, y = Medu, col = sex))
g1 + geom_boxplot() + ylab("Mother's education")
# a plot of high_use and father's education
g2 <- ggplot(alc3, aes(x = high_use, y = Fedu, col = sex))
g2 + geom_boxplot() + ylab("Father's education")
# a plot of high_use and parents' education (Pedu, mean of Medu ja Fedu)
alc3 <- mutate(alc3, Pedu = ((Medu + Fedu) / 2))
g3 <- ggplot(alc3, aes(x = high_use, y = Pedu, col = sex))
g3 + geom_boxplot() + ylab("Parents' education")
5
Use logistic regression to statistically explore the
relationship between your chosen variables and the binary high/low
alcohol consumption variable as the target variable. Present and
interpret a summary of the fitted model. Present and interpret the
coefficients of the model as odds ratios and provide confidence
intervals for them. Interpret the results and compare them to your
previously stated hypothesis.
Hint: If your model includes factor variables, see for example
the RHDS book or the first answer of this stack exchange thread on how R
treats and how you should interpret these variables in the model output
(or use some other resource to study this). (0-5 points)
6
Using the variables which, according to your logistic regression
model, had a statistical relationship with high/low alcohol consumption,
explore the predictive power of you model. Provide a 2x2 cross
tabulation of predictions versus the actual values and optionally
display a graphic visualizing both the actual values and the
predictions. Compute the total proportion of inaccurately classified
individuals (= the training error) and comment on all the results.
Compare the performance of the model with performance achieved by some
simple guessing strategy. (0-3 points)
Bonus: Perform 10-fold cross-validation on your model. Does your
model have better test set performance (smaller prediction error using
10-fold cross-validation) compared to the model introduced in the
Exercise Set (which had about 0.26 error). Could you find such a model?
(0-2 points to compensate any loss of points from the above
exercises)
Super-Bonus: Perform cross-validation to compare the performance
of different logistic regression models (= different sets of
predictors). Start with a very high number of predictors and explore the
changes in the training and testing errors as you move to models with
less predictors. Draw a graph displaying the trends of both training and
testing errors by the number of predictors in the model. (0-4 points to
compensate any loss of points from the above exercises)
After completing all the phases above you are ready to submit
your Assignment for the review (using the Moodle Workshop below). Have
the two links (your GitHub repository and your course diary)
ready!
Assignment 4. Analysis exercises (Max 15 points)
1.The data
Explore the structure and the dimensions of the Boston data and describe the dataset briefly, assuming the reader has no previous knowledge of it. Details about the Boston dataset can be seen for example here. (0-1 points)
The Housing Values in Suburbs of Boston.
Dataset contains 14 colums including crime rate (mean 3.6, median 0.3), pupil-teacher ratio (mean 18.46, median 19.05) and non-retail business acres (mean 408, median 330) per town. Other interesting columns are for example distance from Boston center (mean 3.795, median 3.207), property tax-rate (mean 408.2, median 330) and amount of population of lower status (mean 12.65, median 11.36). There is no missingness in the data and rows 506 (towns?) in the data. All the data contain numerical variables, chas is binary.
Full list of columns are following (from here).
crim per capita crime rate by town.
zn proportion of residential land zoned for lots over 25,000 sq.ft.
indus proportion of non-retail business acres per town.
chas Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).
nox nitrogen oxides concentration (parts per 10 million).
rm average number of rooms per dwelling.
age proportion of owner-occupied units built prior to 1940.
dis weighted mean of distances to five Boston employment centres.
rad index of accessibility to radial highways.
tax full-value property-tax rate per $10,000.
ptratio pupil-teacher ratio by town.
black 1000(Bk−0.63)21000(Bk−0.63)2 where BkBk is the proportion of blacks by town.
lstat lower status of the population (percent).
medv median value of owner-occupied homes in $1000s.
#install.packages("MASS")
library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectlibrary(finalfit)
library(dplyr)
library(corrplot)## corrplot 0.92 loadeddata("Boston")
glimpse(Boston)## Rows: 506
## Columns: 14
## $ crim <dbl> 0.00632, 0.02731, 0.02729, 0.03237, 0.06905, 0.02985, 0.08829,…
## $ zn <dbl> 18.0, 0.0, 0.0, 0.0, 0.0, 0.0, 12.5, 12.5, 12.5, 12.5, 12.5, 1…
## $ indus <dbl> 2.31, 7.07, 7.07, 2.18, 2.18, 2.18, 7.87, 7.87, 7.87, 7.87, 7.…
## $ chas <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ nox <dbl> 0.538, 0.469, 0.469, 0.458, 0.458, 0.458, 0.524, 0.524, 0.524,…
## $ rm <dbl> 6.575, 6.421, 7.185, 6.998, 7.147, 6.430, 6.012, 6.172, 5.631,…
## $ age <dbl> 65.2, 78.9, 61.1, 45.8, 54.2, 58.7, 66.6, 96.1, 100.0, 85.9, 9…
## $ dis <dbl> 4.0900, 4.9671, 4.9671, 6.0622, 6.0622, 6.0622, 5.5605, 5.9505…
## $ rad <int> 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4,…
## $ tax <dbl> 296, 242, 242, 222, 222, 222, 311, 311, 311, 311, 311, 311, 31…
## $ ptratio <dbl> 15.3, 17.8, 17.8, 18.7, 18.7, 18.7, 15.2, 15.2, 15.2, 15.2, 15…
## $ black <dbl> 396.90, 396.90, 392.83, 394.63, 396.90, 394.12, 395.60, 396.90…
## $ lstat <dbl> 4.98, 9.14, 4.03, 2.94, 5.33, 5.21, 12.43, 19.15, 29.93, 17.10…
## $ medv <dbl> 24.0, 21.6, 34.7, 33.4, 36.2, 28.7, 22.9, 27.1, 16.5, 18.9, 15…ff_glimpse(Boston)## $Continuous
## label var_type n missing_n missing_percent mean sd min
## crim crim <dbl> 506 0 0.0 3.6 8.6 0.0
## zn zn <dbl> 506 0 0.0 11.4 23.3 0.0
## indus indus <dbl> 506 0 0.0 11.1 6.9 0.5
## chas chas <int> 506 0 0.0 0.1 0.3 0.0
## nox nox <dbl> 506 0 0.0 0.6 0.1 0.4
## rm rm <dbl> 506 0 0.0 6.3 0.7 3.6
## age age <dbl> 506 0 0.0 68.6 28.1 2.9
## dis dis <dbl> 506 0 0.0 3.8 2.1 1.1
## rad rad <int> 506 0 0.0 9.5 8.7 1.0
## tax tax <dbl> 506 0 0.0 408.2 168.5 187.0
## ptratio ptratio <dbl> 506 0 0.0 18.5 2.2 12.6
## black black <dbl> 506 0 0.0 356.7 91.3 0.3
## lstat lstat <dbl> 506 0 0.0 12.7 7.1 1.7
## medv medv <dbl> 506 0 0.0 22.5 9.2 5.0
## quartile_25 median quartile_75 max
## crim 0.1 0.3 3.7 89.0
## zn 0.0 0.0 12.5 100.0
## indus 5.2 9.7 18.1 27.7
## chas 0.0 0.0 0.0 1.0
## nox 0.4 0.5 0.6 0.9
## rm 5.9 6.2 6.6 8.8
## age 45.0 77.5 94.1 100.0
## dis 2.1 3.2 5.2 12.1
## rad 4.0 5.0 24.0 24.0
## tax 279.0 330.0 666.0 711.0
## ptratio 17.4 19.1 20.2 22.0
## black 375.4 391.4 396.2 396.9
## lstat 6.9 11.4 17.0 38.0
## medv 17.0 21.2 25.0 50.0
##
## $Categorical
## data frame with 0 columns and 506 rowssummary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.002. Graphical overview
Show a graphical overview of the data and show summaries of the variables in the data. Describe and interpret the outputs, commenting on the distributions of the variables and the relationships between them. (0-2 points)
Ggpairs. First I tried to the ggpairs plot matrix from previous weeks. The image is so bad that I had hard time to figuring out whats going on there. I added “proportions = auto” which helped little. Ggpairs show interesting correlations. For example variables listed there is positive correlation between crime rate and low status population ratio and negative correlation between crime rate and population of afro-americans.
Pairs. This plot gives same information than the last one. Ggpairs is bit more helpful when there is the correlation indicator.
Corrplot. This is gives same information than Ggpairs. Probably I will use corrplot in future than ggpairs because this is more tidy and it is easier to see which variables have higher correlation. For example dis (distanse from center) has negative correlation between indus (proportion of non-retail business acres), nox (itrogen oxides concentration) and age ( proportion of owner-occupied units built prior to 1940). Crime rate has positive correlations with rad (index of accessibility to radial highways) and tax (full-value property-tax rate per $10,000).
library(GGally)
library(ggplot2)
# a plot matrix with ggpairs()
p2 <- ggpairs(Boston, mapping = aes(alpha = 0.3), lower = list(combo = wrap("facethist", bins = 20)), proportions = "auto")
p2
pairs(Boston)
# calculate the correlation matrix and round it
cor_matrix <- cor(Boston)
# print the correlation matrix
cor_matrix## crim zn indus chas nox
## crim 1.00000000 -0.20046922 0.40658341 -0.055891582 0.42097171
## zn -0.20046922 1.00000000 -0.53382819 -0.042696719 -0.51660371
## indus 0.40658341 -0.53382819 1.00000000 0.062938027 0.76365145
## chas -0.05589158 -0.04269672 0.06293803 1.000000000 0.09120281
## nox 0.42097171 -0.51660371 0.76365145 0.091202807 1.00000000
## rm -0.21924670 0.31199059 -0.39167585 0.091251225 -0.30218819
## age 0.35273425 -0.56953734 0.64477851 0.086517774 0.73147010
## dis -0.37967009 0.66440822 -0.70802699 -0.099175780 -0.76923011
## rad 0.62550515 -0.31194783 0.59512927 -0.007368241 0.61144056
## tax 0.58276431 -0.31456332 0.72076018 -0.035586518 0.66802320
## ptratio 0.28994558 -0.39167855 0.38324756 -0.121515174 0.18893268
## black -0.38506394 0.17552032 -0.35697654 0.048788485 -0.38005064
## lstat 0.45562148 -0.41299457 0.60379972 -0.053929298 0.59087892
## medv -0.38830461 0.36044534 -0.48372516 0.175260177 -0.42732077
## rm age dis rad tax ptratio
## crim -0.21924670 0.35273425 -0.37967009 0.625505145 0.58276431 0.2899456
## zn 0.31199059 -0.56953734 0.66440822 -0.311947826 -0.31456332 -0.3916785
## indus -0.39167585 0.64477851 -0.70802699 0.595129275 0.72076018 0.3832476
## chas 0.09125123 0.08651777 -0.09917578 -0.007368241 -0.03558652 -0.1215152
## nox -0.30218819 0.73147010 -0.76923011 0.611440563 0.66802320 0.1889327
## rm 1.00000000 -0.24026493 0.20524621 -0.209846668 -0.29204783 -0.3555015
## age -0.24026493 1.00000000 -0.74788054 0.456022452 0.50645559 0.2615150
## dis 0.20524621 -0.74788054 1.00000000 -0.494587930 -0.53443158 -0.2324705
## rad -0.20984667 0.45602245 -0.49458793 1.000000000 0.91022819 0.4647412
## tax -0.29204783 0.50645559 -0.53443158 0.910228189 1.00000000 0.4608530
## ptratio -0.35550149 0.26151501 -0.23247054 0.464741179 0.46085304 1.0000000
## black 0.12806864 -0.27353398 0.29151167 -0.444412816 -0.44180801 -0.1773833
## lstat -0.61380827 0.60233853 -0.49699583 0.488676335 0.54399341 0.3740443
## medv 0.69535995 -0.37695457 0.24992873 -0.381626231 -0.46853593 -0.5077867
## black lstat medv
## crim -0.38506394 0.4556215 -0.3883046
## zn 0.17552032 -0.4129946 0.3604453
## indus -0.35697654 0.6037997 -0.4837252
## chas 0.04878848 -0.0539293 0.1752602
## nox -0.38005064 0.5908789 -0.4273208
## rm 0.12806864 -0.6138083 0.6953599
## age -0.27353398 0.6023385 -0.3769546
## dis 0.29151167 -0.4969958 0.2499287
## rad -0.44441282 0.4886763 -0.3816262
## tax -0.44180801 0.5439934 -0.4685359
## ptratio -0.17738330 0.3740443 -0.5077867
## black 1.00000000 -0.3660869 0.3334608
## lstat -0.36608690 1.0000000 -0.7376627
## medv 0.33346082 -0.7376627 1.0000000# visualize the correlation matrix
library(corrplot)
corrplot(cor_matrix, method="circle")
3. Standardize the dataset
Standardize the dataset and print out summaries of the scaled data. How did the variables change? Create a categorical variable of the crime rate in the Boston dataset (from the scaled crime rate). Use the quantiles as the break points in the categorical variable. Drop the old crime rate variable from the dataset. Divide the dataset to train and test sets, so that 80% of the data belongs to the train set. (0-2 points)
Standardize & scale the dataset. How did the variables change? First I notice that crime rate dropped from mean 3.61 and median 0.25651 to mean 0 and median -0.390280. Max values in every variable dropped significantly.
I tried corrplot out of curiosity and see no changes (no ****, sherlock).
# center and standardize variables
boston_scaled <- as.data.frame(scale(Boston))
boston_scaled$crim <- as.numeric(boston_scaled$crim)
# summaries of the scaled variables
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865glimpse(boston_scaled)## Rows: 506
## Columns: 14
## $ crim <dbl> -0.4193669, -0.4169267, -0.4169290, -0.4163384, -0.4120741, -0…
## $ zn <dbl> 0.28454827, -0.48724019, -0.48724019, -0.48724019, -0.48724019…
## $ indus <dbl> -1.2866362, -0.5927944, -0.5927944, -1.3055857, -1.3055857, -1…
## $ chas <dbl> -0.2723291, -0.2723291, -0.2723291, -0.2723291, -0.2723291, -0…
## $ nox <dbl> -0.1440749, -0.7395304, -0.7395304, -0.8344581, -0.8344581, -0…
## $ rm <dbl> 0.4132629, 0.1940824, 1.2814456, 1.0152978, 1.2273620, 0.20689…
## $ age <dbl> -0.11989477, 0.36680343, -0.26554897, -0.80908783, -0.51067434…
## $ dis <dbl> 0.1400749840, 0.5566090496, 0.5566090496, 1.0766711351, 1.0766…
## $ rad <dbl> -0.9818712, -0.8670245, -0.8670245, -0.7521778, -0.7521778, -0…
## $ tax <dbl> -0.6659492, -0.9863534, -0.9863534, -1.1050216, -1.1050216, -1…
## $ ptratio <dbl> -1.4575580, -0.3027945, -0.3027945, 0.1129203, 0.1129203, 0.11…
## $ black <dbl> 0.4406159, 0.4406159, 0.3960351, 0.4157514, 0.4406159, 0.41016…
## $ lstat <dbl> -1.07449897, -0.49195252, -1.20753241, -1.36017078, -1.0254866…
## $ medv <dbl> 0.15952779, -0.10142392, 1.32293748, 1.18158864, 1.48603229, 0…#cor_matrix2 <- cor(boston_scaled)
#cor_matrix2
#corrplot(cor_matrix2, method="circle")Create a categorical variable of the crime rate in the Boston dataset (from the scaled crime rate). Use the quantiles as the break points in the categorical variable.
bins <- quantile(boston_scaled$crim)
bins## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610# created a categorical variable 'crime': low, med_low, med_high and high
crime <- cut(boston_scaled$crim, breaks = bins, labels = c("low", "med_low", "med_high", "high"), include.lowest = TRUE)
# looking at the table of the new factor crime
crime## [1] low low low low low low med_low med_low
## [9] med_low med_low med_low med_low med_low med_high med_high med_high
## [17] med_high med_high med_high med_high med_high med_high med_high med_high
## [25] med_high med_high med_high med_high med_high med_high med_high med_high
## [33] med_high med_high med_high low med_low low med_low low
## [41] low med_low med_low med_low med_low med_low med_low med_low
## [49] med_low med_low med_low low low low low low
## [57] low low med_low med_low med_low med_low med_low med_low
## [65] low low low low med_low med_low med_low med_low
## [73] med_low med_low low med_low med_low med_low low med_low
## [81] low low low low low low low low
## [89] low low low low low low low med_low
## [97] med_low med_low low low med_low med_low med_low med_low
## [105] med_low med_low med_low med_low med_low med_high med_low med_low
## [113] med_low med_low med_low med_low med_low med_low med_low med_low
## [121] low low med_low med_low med_low med_low med_high med_high
## [129] med_high med_high med_high med_high med_high med_high med_high med_high
## [137] med_high med_high med_low med_high med_high med_high med_high high
## [145] med_high med_high med_high med_high med_high med_high med_high med_high
## [153] med_high med_high med_high med_high med_high med_high med_high med_high
## [161] med_high med_high med_high med_high med_high med_high med_high med_high
## [169] med_high med_high med_high med_high med_low med_low med_low low
## [177] low low low low low low med_low med_low
## [185] med_low low low low med_low med_low med_low low
## [193] med_low low low low low low low low
## [201] low low low low low med_low med_low med_low
## [209] med_low med_high med_low med_high med_low med_low med_high med_low
## [217] low low med_low med_low med_high med_high med_high med_high
## [225] med_high med_high med_high med_high med_high med_high med_high med_high
## [233] med_high med_high med_high med_high med_high med_high med_low med_low
## [241] med_low med_low med_low med_low med_low med_low med_high med_low
## [249] med_low med_low med_low med_low med_low med_high low low
## [257] low med_high med_high med_high med_high med_high med_high med_high
## [265] med_high med_high med_high med_high med_high med_low med_high med_low
## [273] med_low med_low low med_low med_low low low med_low
## [281] low low low low low low low low
## [289] low low low low low med_low low med_low
## [297] low med_low low low low low med_low med_low
## [305] low low low low med_high med_high med_high med_high
## [313] med_high med_high med_high med_low med_high med_low med_high med_high
## [321] med_low med_low med_high med_high med_high med_low med_high med_low
## [329] low low low low low low low low
## [337] low low low low low low low low
## [345] low low low low low low low low
## [353] low low low med_low high high high high
## [361] high high high high med_high high high high
## [369] high high high high high high high high
## [377] high high high high high high high high
## [385] high high high high high high high high
## [393] high high high high high high high high
## [401] high high high high high high high high
## [409] high high high high high high high high
## [417] high high high high high high high high
## [425] high high high high high high high high
## [433] high high high high high high high high
## [441] high high high high high high high high
## [449] high high high high high high high high
## [457] high high high high high high high high
## [465] high med_high high high high high high high
## [473] med_high high high high high high high high
## [481] high high high med_high med_high med_high high high
## [489] med_low med_low med_low med_low med_low med_low med_high med_low
## [497] med_high med_high med_low med_low med_low low low low
## [505] med_low low
## Levels: low med_low med_high highDrop the old crime rate variable from the dataset
# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)
# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)
summary(boston_scaled)## zn indus chas nox
## Min. :-0.48724 Min. :-1.5563 Min. :-0.2723 Min. :-1.4644
## 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723 1st Qu.:-0.9121
## Median :-0.48724 Median :-0.2109 Median :-0.2723 Median :-0.1441
## Mean : 0.00000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723 3rd Qu.: 0.5981
## Max. : 3.80047 Max. : 2.4202 Max. : 3.6648 Max. : 2.7296
## rm age dis rad
## Min. :-3.8764 Min. :-2.3331 Min. :-1.2658 Min. :-0.9819
## 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049 1st Qu.:-0.6373
## Median :-0.1084 Median : 0.3171 Median :-0.2790 Median :-0.5225
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617 3rd Qu.: 1.6596
## Max. : 3.5515 Max. : 1.1164 Max. : 3.9566 Max. : 1.6596
## tax ptratio black lstat
## Min. :-1.3127 Min. :-2.7047 Min. :-3.9033 Min. :-1.5296
## 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049 1st Qu.:-0.7986
## Median :-0.4642 Median : 0.2746 Median : 0.3808 Median :-0.1811
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332 3rd Qu.: 0.6024
## Max. : 1.7964 Max. : 1.6372 Max. : 0.4406 Max. : 3.5453
## medv crime
## Min. :-1.9063 low :127
## 1st Qu.:-0.5989 med_low :126
## Median :-0.1449 med_high:126
## Mean : 0.0000 high :127
## 3rd Qu.: 0.2683
## Max. : 2.9865Divide the dataset to train and test sets, so that 80% of the data belongs to the train set.
# number of rows in the Boston dataset
n <- nrow(boston_scaled)
# choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
# create train set
train <- boston_scaled[ind,]
# create test set
test <- boston_scaled[-ind,]4. Linear discriminant analysis
Fit the linear discriminant analysis on the train set. Use the categorical crime rate as the target variable and all the other variables in the dataset as predictor variables. Draw the LDA (bi)plot. (0-3 points)
# linear discriminant analysis
lda.fit <- lda(crime ~ ., data = train)
# print the lda.fit object
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2599010 0.2549505 0.2376238 0.2475248
##
## Group means:
## zn indus chas nox rm age
## low 0.92362178 -0.9275127 -0.084848104 -0.8769659 0.49097238 -0.8644060
## med_low -0.04389892 -0.2831072 0.033465125 -0.5732268 -0.12841506 -0.3202167
## med_high -0.39612627 0.2289412 0.219808462 0.4391551 -0.02824116 0.3891548
## high -0.48724019 1.0171519 0.003267949 1.1024120 -0.32879128 0.7919005
## dis rad tax ptratio black lstat
## low 0.8666169 -0.6953013 -0.7258484 -0.48139792 0.37684973 -0.77312988
## med_low 0.3836171 -0.5481298 -0.4775778 -0.08439846 0.30723713 -0.11020025
## med_high -0.3637747 -0.4016561 -0.2849129 -0.25949087 0.08429679 0.07675086
## high -0.8311418 1.6377820 1.5138081 0.78037363 -0.72352297 0.82352417
## medv
## low 0.539668150
## med_low -0.006206102
## med_high 0.053969199
## high -0.616803536
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.137407634 0.695640215 -0.81206177
## indus -0.008039677 -0.353190755 0.44397481
## chas -0.082181368 -0.044875553 0.13762568
## nox 0.394158829 -0.719478139 -1.47250904
## rm -0.086926072 -0.017940781 -0.24430079
## age 0.277889142 -0.245164029 -0.01584747
## dis -0.087752386 -0.295594127 0.22891573
## rad 3.199869562 0.855247129 0.09667593
## tax -0.015845123 0.130055486 0.41803838
## ptratio 0.149849473 0.030245539 -0.26752203
## black -0.095871469 0.004824314 0.11359994
## lstat 0.235174302 -0.207202173 0.36872324
## medv 0.198368301 -0.332534641 -0.15186908
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9533 0.0346 0.0121# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
graphics::arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(train$crime)
# plot the lda results (select both lines and execute them at the same time!)
plot(lda.fit, dimen = 2)
lda.arrows(lda.fit, myscale = 1)
5. Predict & cross tabulate
Save the crime categories from the test set and then remove the categorical crime variable from the test dataset. Then predict the classes with the LDA model on the test data. Cross tabulate the results with the crime categories from the test set. Comment on the results. (0-3 points)
correct_classes <- test$crime
test <- dplyr::select(test, -crime)
# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 14 7 1 0
## med_low 2 19 2 0
## med_high 0 11 18 1
## high 0 0 0 276. Standardize the dataset
Reload the Boston dataset and standardize the dataset (we did not do this in the Exercise Set, but you should scale the variables to get comparable distances).
data("Boston")
# center and standardize variables
boston_scaled <- scale(Boston)
# summaries of the scaled variables
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865# class of the boston_scaled object
class(boston_scaled)## [1] "matrix" "array"# change the object to data frame
boston_scaled <- as.data.frame(boston_scaled)Calculate the distances between the observations.
# euclidean distance matrix
dist_eu <- dist(Boston)
# look at the summary of the distances
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.119 85.624 170.539 226.315 371.950 626.047# manhattan distance matrix
dist_man <- dist(Boston, method = "manhattan")
# look at the summary of the distances
summary(dist_man)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 2.016 149.145 279.505 342.899 509.707 1198.265Run k-means algorithm on the dataset.
km <- kmeans(Boston, centers = 4)
# plot the Boston dataset with clusters
pairs(Boston, col = km$cluster)
# k-means clustering
km <- kmeans(Boston, centers = 4)
# plot the Boston dataset with clusters
pairs(Boston[6:10], col = km$cluster)
####
# k-means clustering
km <- kmeans(Boston, centers = 3)
# plot the Boston dataset with clusters
pairs(Boston[c("rm", "age", "dis", "crim")], col = km$cluster)
Investigate what is the optimal number of clusters and run the algorithm again.
set.seed(123)
# determine the number of clusters
k_max <- 10
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')## Warning: `qplot()` was deprecated in ggplot2 3.4.0.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
# k-means clustering
km <- kmeans(Boston, centers = 10)
# plot the Boston dataset with clusters
pairs(Boston, col = km$cluster)
The optimal number of clusters is when the total WCSS drops radically. In this example twcss drops when amount of clusters is two. I run the algorithm again with this.
k_max <- 2
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})Visualize the clusters (for example with the pairs() or ggpairs() functions, where the clusters are separated with colors) and interpret the results. (0-4 points)
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')
# k-means clustering
km <- kmeans(Boston, centers = 2)
# plot the Boston dataset with clusters
pairs(Boston, col = km$cluster)
(more chapters to be added similarly as we proceed with the course!)